To find \( (fg)(x) \), we need to multiply the two functions \( f(x) \) and \( g(x) \).
Given: \[ f(x) = 2x^2 - x - 6 \] \[ g(x) = x^2 + 3x - 10 \]
Calculating \( (fg)(x) \): \[ (fg)(x) = f(x) \cdot g(x) = (2x^2 - x - 6)(x^2 + 3x - 10) \]
Now let's perform the multiplication:
Expanding: \[ (2x^2 - x - 6)(x^2 + 3x - 10) = 2x^2(x^2) + 2x^2(3x) + 2x^2(-10) - x(x^2) - x(3x) - x(-10) - 6(x^2) - 6(3x) - 6(-10) \]
Calculating term by term: \[ = 2x^4 + 6x^3 - 20x^2 - x^3 - 3x^2 + 10x - 6x^2 - 18x + 60 \]
Now, combine like terms: \[ = 2x^4 + (6x^3 - x^3) + (-20x^2 - 3x^2 - 6x^2) + (10x - 18x) + 60 \] \[ = 2x^4 + 5x^3 - 29x^2 - 8x + 60 \]
Thus, \[ (fg)(x) = 2x^4 + 5x^3 - 29x^2 - 8x + 60 \]
Finding the Domain of \( (fg)(x) \)
Both \( f(x) \) and \( g(x) \) are polynomials, and the domain of a polynomial function is always all real numbers. Therefore, the domain of \( (fg)(x) \) is also all real numbers.
Final Answers:
- \( (fg)(x) = 2x^4 + 5x^3 - 29x^2 - 8x + 60 \)
- Domain of \( (fg)(x) \): all real numbers, \( (-\infty, \infty) \).